Showing 1–50 of 434 results for author: Zhou, X

The Minimum Number of Edges in $(p+1)K_2$-Saturated Graphs

Authors: Xiaoteng Zhou, Kazuya Haraguchi, Hanchun Yuan

Abstract: Given a family of graphs $\mathcal{F}$, a graph $G$ is $\mathcal{F}$-saturated if it is $\mathcal{F}$-free but the addition of any missing edge creates a copy of some $F \in \mathcal{F}$. The study of the minimum number of edges in $\mathcal{F}$-saturated graphs is a central topic in extremal graph theory. Let $(p+1)K_2$ denote a matching of size $p+1$. Determining the minimum number of edges in… ▽ More Given a family of graphs $\mathcal{F}$, a graph $G$ is $\mathcal{F}$-saturated if it is $\mathcal{F}$-free but the addition of any missing edge creates a copy of some $F \in \mathcal{F}$. The study of the minimum number of edges in $\mathcal{F}$-saturated graphs is a central topic in extremal graph theory. Let $(p+1)K_2$ denote a matching of size $p+1$. Determining the minimum number of edges in a $(p+1)K_{2}$-saturated graph is a fundamental question in this area, explicitly posed as Problem 9 in the survey by Faudree et al. (2011). In this paper, we refine the structural analysis of $(p+1)K_2$-saturated graphs and derive an explicit formula for the number of edges in terms of a single integer parameter. By minimizing this formula we determine $\mathrm{sat}(n,(p+1)K_2)$ for all $n>2p$, thereby resolving Problem 9 in full generality and extending earlier results of Kászonyi--Tuza (1986) and Zhang--Lu--Yu (2024). Moreover, by maximizing the same formula we recover the classical Erdős--Gallai (1959) upper bound on the number of edges in such graphs. △ Less

Submitted 16 November, 2025; originally announced November 2025.

MSC Class: 05C35; 05C70

Pairs of Embedded Spheres with Pinched Prescribed Mean Curvature

Authors: Liam Mazurowski, Xin Zhou

Abstract: Assume $h$ is a positive function on the unit three-sphere which satisfies the pinching condition $h < h_0 \approx 0.547$. We prove the existence of at least two embedded two-spheres with prescribed mean curvature $h$. The same result holds for sign-changing functions $h$ satisfying $\vert h\vert < h_0$ under a mild assumption on the zero set. Assume $h$ is a positive function on the unit three-sphere which satisfies the pinching condition $h < h_0 \approx 0.547$. We prove the existence of at least two embedded two-spheres with prescribed mean curvature $h$. The same result holds for sign-changing functions $h$ satisfying $\vert h\vert < h_0$ under a mild assumption on the zero set. △ Less

Submitted 11 November, 2025; originally announced November 2025.

Comments: 15 pages, comments welcome!

MSC Class: 53A10

Geometric inequalities related to fractional perimeter: fractional Poincaré, isoperimetric, and boxing inequalities in metric measure spaces

Authors: Josh Kline, Panu Lahti, Jiang Li, Xiaodan Zhou

Abstract: In the setting of a complete, doubling metric measure space $(X,d,μ)$ supporting a $(1,1)$-Poincaré inequality, we show that for all $0<θ<1$, the following fractional Poincaré inequality holds for all balls $B$ and locally integrable functions $u$, $$ \int_{B}|u-u_B|dμ\le C(1-θ)\,\text{rad}(B)^θ\int_{τB}\int_{τB}\frac{|u(x)-u(y)|}{d(x,y)^θμ(B(x,d(x,y)))}dμ(y)dμ(x), $$ where $C\ge 1$ and… ▽ More In the setting of a complete, doubling metric measure space $(X,d,μ)$ supporting a $(1,1)$-Poincaré inequality, we show that for all $0<θ<1$, the following fractional Poincaré inequality holds for all balls $B$ and locally integrable functions $u$, $$ \int_{B}|u-u_B|dμ\le C(1-θ)\,\text{rad}(B)^θ\int_{τB}\int_{τB}\frac{|u(x)-u(y)|}{d(x,y)^θμ(B(x,d(x,y)))}dμ(y)dμ(x), $$ where $C\ge 1$ and $τ\ge 1$ are constants depending only on the doubling and $(1,1)$-Poincaré inequality constants. Notably, this inequality features the scaling constant $(1-θ)$ present in the Bourgain-Brezis-Mironescu theory characterizing Sobolev functions via nonlocal functionals. From this inequality, we obtain a fractional relative isoperimetric inequality as well as global and local versions of a fractional boxing inequality, each featuring the same scaling constant $(1-θ)$ and defined in terms of the fractional $θ$-perimeter, and prove equivalences with the above fractional Poincaré inequality. We also show that $(X,d,μ)$ supports a $(1,1)$-Poincaré inequality if and only if the above fractional Poincaré inequality holds for all $θ$ sufficiently close to $1$. Under the additional assumption of lower Ahlfors $Q$-regularity of the measure $μ$, we additionally use the aforementioned results to establish global inequalities, in the form of fractional isoperimetric and fractional Sobolev inequalities, which also feature the scaling constant $(1-θ)$. Moreover, we prove that such inequalities are equivalent with the lower Ahlfors $Q$-regularity condition on the measure. △ Less

Submitted 6 November, 2025; originally announced November 2025.

Comments: 54 pages, 1 figure

MSC Class: 30L15 46E36

On Kodaira dimension and scalar curvature in almost Hermitian geometry

Authors: Xianchao Zhou

Abstract: In this paper, we investigate Riemannian curvature constraints on the Kodaira dimension of compact almost Hermitian manifolds. Specifically, for a compact almost Hermitian manifold $(M, J, g)$ in the Gray-Hervella class $\mathcal{W}_2\oplus\mathcal{W}_3\oplus \mathcal{W}_4$ with nonnegative Riemannian scalar curvature, we prove that its Kodaira dimension must satisfy $κ(M, J)=-\infty$; or… ▽ More In this paper, we investigate Riemannian curvature constraints on the Kodaira dimension of compact almost Hermitian manifolds. Specifically, for a compact almost Hermitian manifold $(M, J, g)$ in the Gray-Hervella class $\mathcal{W}_2\oplus\mathcal{W}_3\oplus \mathcal{W}_4$ with nonnegative Riemannian scalar curvature, we prove that its Kodaira dimension must satisfy $κ(M, J)=-\infty$; or $κ(M, J)=0$, in which case $(M,J,g)$ is a Kähler Calabi-Yau manifold. The same conclusions also hold for compact Hermitian manifolds with an assumption of nonnegative mixed scalar curvature. As an important example, we study the twistor geometry of a compact anti-self-dual 4-manifold. In particular, for the twistor space with the Eells-Salamon almost complex structure, we show that the Kodaira dimension is zero. △ Less

Submitted 19 October, 2025; originally announced October 2025.

Comments: 20 pages. Comments are welcome!

Macaulay representation of the prolongation matrix and the SOS conjecture

Authors: Zhiwei Wang, Chenlong Yue, Xiangyu Zhou

Abstract: Let $z \in \mathbb{C}^n$, and let $A(z,\bar{z})$ be a real valued diagonal bihomogeneous Hermitian polynomial such that $A(z,\bar{z})\|z\|^2$ is a sum of squares, where $\|z\|$ denotes the Euclidean norm of $z$. In this paper, we provide an estimate for the rank of the sum of squares $A(z,\bar{z})\|z\|^2$ when $A(z,\bar{z})$ is not semipositive definite. As a consequence, we confirm the SOS conjec… ▽ More Let $z \in \mathbb{C}^n$, and let $A(z,\bar{z})$ be a real valued diagonal bihomogeneous Hermitian polynomial such that $A(z,\bar{z})\|z\|^2$ is a sum of squares, where $\|z\|$ denotes the Euclidean norm of $z$. In this paper, we provide an estimate for the rank of the sum of squares $A(z,\bar{z})\|z\|^2$ when $A(z,\bar{z})$ is not semipositive definite. As a consequence, we confirm the SOS conjecture proposed by Ebenfelt for $2 \leq n \leq 6$ when $A(z,\bar{z})$ is a real valued diagonal (not necessarily bihomogeneous) Hermitian polynomial, and we also give partial answers to the SOS conjecture for $n\geq 7$. △ Less

Submitted 5 September, 2025; v1 submitted 4 September, 2025; originally announced September 2025.

Comments: Comments welcome! 22pages

Morse Index Classification and Landscape of Kuramoto System for Hebbian-based Binary Pattern Recognition

Authors: Xiaoxue Zhao, Xiang Zhou

Abstract: This study examines the Kuramoto model with a Hebbian learning rule and second-order Fourier coupling for binary pattern recognition. The system stores memorized binary patterns as stable critical points, enabling it to identify the closest match to a defective input. However, the system exhibits multiple stable states and thus the dynamics are influenced by saddle points and other unstable critic… ▽ More This study examines the Kuramoto model with a Hebbian learning rule and second-order Fourier coupling for binary pattern recognition. The system stores memorized binary patterns as stable critical points, enabling it to identify the closest match to a defective input. However, the system exhibits multiple stable states and thus the dynamics are influenced by saddle points and other unstable critical points, which may disrupt convergence and recognition accuracy. We systematically classify the stability of these critical points by analyzing the Morse index, which quantifies the stability of critical points by the number of unstable directions. The index-1 saddle point is highlighted as the transition state on the energy landscape of the Kuramoto model. These findings provide deeper insights into the stability landscape of the Kuramoto model than the stable equilibria, enhancing its theoretical foundation for binary pattern recognition. △ Less

Submitted 28 August, 2025; originally announced August 2025.

Threshold Diffusions

Authors: Lina Ji, Chuyang Li, Xiaowen Zhou

Abstract: We propose threshold diffusion processes as unique solutions to stochastic differential equations with step-function coefficients, and obtain explicit expressions for the conditional Laplace transform of the hitting times and the potential measures. Applying these results, we further discuss their asymptotic behaviors such as the stationary distributions and the escape probabilities. We propose threshold diffusion processes as unique solutions to stochastic differential equations with step-function coefficients, and obtain explicit expressions for the conditional Laplace transform of the hitting times and the potential measures. Applying these results, we further discuss their asymptotic behaviors such as the stationary distributions and the escape probabilities. △ Less

Submitted 25 August, 2025; originally announced August 2025.

Superposition Property in Disjoint Variables for the Infinity Laplace Equation

Authors: Qing Liu, Juan J. Manfredi, Xiaodan Zhou

Abstract: We establish a superposition principle in disjoint variables for the inhomogeneous infinity-Laplace equation. We show that the sum of viscosity solutions of the inhomogeneous infinity-Laplace equation in separate domains is a viscosity solution in the product domain. This result has been used in the literature with certain particular choices of solutions to simplify regularity analysis for a gener… ▽ More We establish a superposition principle in disjoint variables for the inhomogeneous infinity-Laplace equation. We show that the sum of viscosity solutions of the inhomogeneous infinity-Laplace equation in separate domains is a viscosity solution in the product domain. This result has been used in the literature with certain particular choices of solutions to simplify regularity analysis for a general inhomogeneous infinity-Laplace equation by reducing it to the case without sign-changing inhomogeneous terms and vanishing gradient singularities. We present a proof of this superposition principle for general viscosity solutions. We also explore generalization in metric spaces using cone comparison techniques and study related properties for general elliptic and convex equations. △ Less

Submitted 13 September, 2025; v1 submitted 23 August, 2025; originally announced August 2025.

Comments: 12 pages

MSC Class: 35D40; 35J92

The Liouville-type equation and an Onofri-type inequality on closed 4-manifolds

Authors: Xi-Nan Ma, Tian Wu, Xiao Zhou

Abstract: In this paper, we study the Liouville-type equation \[Δ^2 u-λ_1κΔu+λ_2κ^2(1-\mathrm e^{4u})=0\] on a closed Riemannian manifold \((M^4,g)\) with \(\operatorname{Ric}\geqslant 3κg\) and \(κ>0\). Using the method of invariant tensors, we derive a differential identity to classify solutions within certain ranges of the parameters \(λ_1,λ_2\). A key step in our proof is a second-order derivative e… ▽ More In this paper, we study the Liouville-type equation \[Δ^2 u-λ_1κΔu+λ_2κ^2(1-\mathrm e^{4u})=0\] on a closed Riemannian manifold \((M^4,g)\) with \(\operatorname{Ric}\geqslant 3κg\) and \(κ>0\). Using the method of invariant tensors, we derive a differential identity to classify solutions within certain ranges of the parameters \(λ_1,λ_2\). A key step in our proof is a second-order derivative estimate, which is established via the continuity method. As an application of the classification results, we derive an Onofri-type inequality on the 4-sphere and prove its rigidity. △ Less

Submitted 20 August, 2025; originally announced August 2025.

Pohozaev identities for weak solutions of Grushin type p-sub-Laplacian equation via domain variations

Authors: Yawei Wei, Xiaodong Zhou

Abstract: In this paper, we study Pohozaev identities for weak solutions of degenerate elliptic equations involving Grushin type p-sub-Laplacian under only $C^1$-regularity assumption. By using domain variations, we obtain the local Pohozaev identities of translating type and scaling type. As an application, a global Pohozaev identity of scaling type in $\mathbb{R}^{N+l}$ is also derived. In this paper, we study Pohozaev identities for weak solutions of degenerate elliptic equations involving Grushin type p-sub-Laplacian under only $C^1$-regularity assumption. By using domain variations, we obtain the local Pohozaev identities of translating type and scaling type. As an application, a global Pohozaev identity of scaling type in $\mathbb{R}^{N+l}$ is also derived. △ Less

Submitted 26 July, 2025; originally announced July 2025.

MSC Class: 35A22; 35J20; 35J70

Data-Driven Exploration for a Class of Continuous-Time Indefinite Linear--Quadratic Reinforcement Learning Problems

Authors: Yilie Huang, Xun Yu Zhou

Abstract: We study reinforcement learning (RL) for the same class of continuous-time stochastic linear--quadratic (LQ) control problems as in \cite{huang2024sublinear}, where volatilities depend on both states and controls while states are scalar-valued and running control rewards are absent. We propose a model-free, data-driven exploration mechanism that adaptively adjusts entropy regularization by the cri… ▽ More We study reinforcement learning (RL) for the same class of continuous-time stochastic linear--quadratic (LQ) control problems as in \cite{huang2024sublinear}, where volatilities depend on both states and controls while states are scalar-valued and running control rewards are absent. We propose a model-free, data-driven exploration mechanism that adaptively adjusts entropy regularization by the critic and policy variance by the actor. Unlike the constant or deterministic exploration schedules employed in \cite{huang2024sublinear}, which require extensive tuning for implementations and ignore learning progresses during iterations, our adaptive exploratory approach boosts learning efficiency with minimal tuning. Despite its flexibility, our method achieves a sublinear regret bound that matches the best-known model-free results for this class of LQ problems, which were previously derived only with fixed exploration schedules. Numerical experiments demonstrate that adaptive explorations accelerate convergence and improve regret performance compared to the non-adaptive model-free and model-based counterparts. △ Less

Submitted 23 July, 2025; v1 submitted 30 June, 2025; originally announced July 2025.

Comments: 37 pages, 10 figures

Characterization of negative line bundles whose Grauert blow-down are quadratic transforms

Authors: Fusheng Deng, Yinji Li, Qunhuan Liu, Xiangyu Zhou

Abstract: We show that the Grauert blow-down of a holomorphic negative line bundle $L$ over a compact complex space is a quadratic transform if and only if $k_0L^*$ is very ample and $(k_0+1)L^*$ is globally generated, where $k_0$ is the initial order of $L^*$, namely, the minimal integer such that $k_0^*$ has nontrivial holomorphic section. We show that the Grauert blow-down of a holomorphic negative line bundle $L$ over a compact complex space is a quadratic transform if and only if $k_0L^*$ is very ample and $(k_0+1)L^*$ is globally generated, where $k_0$ is the initial order of $L^*$, namely, the minimal integer such that $k_0^*$ has nontrivial holomorphic section. △ Less

Submitted 17 June, 2025; originally announced June 2025.

Comments: Comments welcome!

The Bellman Function for Level Sets of Sparse Operators

Authors: John Freeland Small, Irina Holmes Fay, Zachary H. Pence, Xiaokun Zhou

Abstract: We investigate weak-type $(1, 1)$ boundedness of sparse operators with respect to Lebesgue measure. Specifically, we find the Bellman function maximizing level sets of sparse operators (localized to an interval) and use this to find the exact weak-$(1,1)$ norm of these sparse operators. We investigate weak-type $(1, 1)$ boundedness of sparse operators with respect to Lebesgue measure. Specifically, we find the Bellman function maximizing level sets of sparse operators (localized to an interval) and use this to find the exact weak-$(1,1)$ norm of these sparse operators. △ Less

Submitted 13 June, 2025; originally announced June 2025.

Homeomorphic Sobolev extensions and integrability of hyperbolic metric

Authors: Xilin Zhou

Abstract: Very recently, it was proved that if the hyperbolic metric of a planar Jordan domain is $L^q$-integrable for some $q\in (1,\infty)$, then every homeomorphic parametrization of the boundary Jordan curve via the unit circle can be extended to a Sobolev homeomorphism of the entire disk. This naturally raises the question of whether the extension holds under more general integrability conditions on… ▽ More Very recently, it was proved that if the hyperbolic metric of a planar Jordan domain is $L^q$-integrable for some $q\in (1,\infty)$, then every homeomorphic parametrization of the boundary Jordan curve via the unit circle can be extended to a Sobolev homeomorphism of the entire disk. This naturally raises the question of whether the extension holds under more general integrability conditions on the hyperbolic metric. In this work, we examine the case where the hyperbolic metric is $φ$-integrable. Under appropriate conditions on the function $φ$, we establish the existence of a Sobolev homeomorphic extension for every homeomorphic parametrization of the Jordan curve. Moreover, we demonstrate the sharpness of our result by providing an explicit counterexample. △ Less

Submitted 11 June, 2025; originally announced June 2025.

MSC Class: 46E35; 30C62; 58E20

Coupling of forward-backward stochastic differential equations on the Wiener space, and application on regularity

Authors: Xilin Zhou

Abstract: S. Geiss and J. Ylinen proposed the coupling method \cite{Geiss:Ylinen:21} to investigate the regularity for the solution to the backward stochastic differential equations with random coefficients. In this paper, we explore this method in setting for the forward-backward stochastic differential equation with random and Lipschitz coefficients, We obtain the regularity in time, and the Malliavin Sob… ▽ More S. Geiss and J. Ylinen proposed the coupling method \cite{Geiss:Ylinen:21} to investigate the regularity for the solution to the backward stochastic differential equations with random coefficients. In this paper, we explore this method in setting for the forward-backward stochastic differential equation with random and Lipschitz coefficients, We obtain the regularity in time, and the Malliavin Sobolev ${\mathbb D}_{1,2}$ differentiability for the solution. △ Less

Submitted 11 June, 2025; originally announced June 2025.

MSC Class: 60H07; 60H10; 46E35

Transition Path Theory For Lévy-Type Processes: SDE Representation and Statistics

Authors: Yuanfei Huang, Xiang Zhou

Abstract: This paper establishes a Transition Path Theory (TPT) for Lévy-type processes, addressing a critical gap in the study of the transition mechanism between meta-stabile states in non-Gaussian stochastic systems. A key contribution is the rigorous derivation of the stochastic differential equation (SDE) representation for transition path processes, which share the same distributional properties as tr… ▽ More This paper establishes a Transition Path Theory (TPT) for Lévy-type processes, addressing a critical gap in the study of the transition mechanism between meta-stabile states in non-Gaussian stochastic systems. A key contribution is the rigorous derivation of the stochastic differential equation (SDE) representation for transition path processes, which share the same distributional properties as transition trajectories, along with a proof of its well-posedness. This result provides a solid theoretical foundation for sampling transition trajectories. The paper also investigates the statistical properties of transition trajectories, including their probability distribution, probability current, and rate of occurrence. △ Less

Submitted 11 June, 2025; originally announced June 2025.

Speed of coming down from infinity for $Λ$-Fleming-Viot initial support

Authors: Huili Liu, Xiaowen Zhou

Abstract: The $Λ$-Fleming-Viot process is a probability measure-valued process that is dual to a $Λ$-coalescent that allows multiple collisions. In this paper, we consider a class of $Λ$-Fleming-Viot processes with Brownian spatial motion and with associated $Λ$-coalescents that come down from infinity. Notably, these processes have the compact support property: the support of the process becomes finite as… ▽ More The $Λ$-Fleming-Viot process is a probability measure-valued process that is dual to a $Λ$-coalescent that allows multiple collisions. In this paper, we consider a class of $Λ$-Fleming-Viot processes with Brownian spatial motion and with associated $Λ$-coalescents that come down from infinity. Notably, these processes have the compact support property: the support of the process becomes finite as soon as $t>0$, even though the initial measure has unbounded support. We obtain asymptotic results characterizing the rates at which the initial supports become finite. The rates of coming down are expressed in terms of the asymptotic inverse function of the tail distribution of the initial measure and the speed function of coming down from infinity for the corresponding $Λ$-coalescent. △ Less

Submitted 8 June, 2025; originally announced June 2025.

Comments: 28 pages

General monotone formula for homogeneous $k$-Hessian equation in the exterior domain and its applications

Authors: Jiabin Yin, Xingjian Zhou

Abstract: In this paper, we deal with an overdetermined problem for the $k$-Hessian equation ($1\leq k<\frac n2$) in the exterior domain and prove the corresponding ball characterizations. Since that Weinberger type approach seems to fail to solve the problem, we give a new perspective to solve exterior overdetermined problem by combining two integral identities and geometric inequalities inspired by Brando… ▽ More In this paper, we deal with an overdetermined problem for the $k$-Hessian equation ($1\leq k<\frac n2$) in the exterior domain and prove the corresponding ball characterizations. Since that Weinberger type approach seems to fail to solve the problem, we give a new perspective to solve exterior overdetermined problem by combining two integral identities and geometric inequalities inspired by Brandolini-Nitsch-Salani's results \cite{BNS}. Meanwhile, we establish general monotone formulas to derive geometric inequalities related to $k$-admissible solution $u$ in $\mathbb R^n\setminusΩ$, where $Ω$ is smooth, $k$-convex and star-shaped domain, which constructed by Ma-Zhang\cite{MZ} and Xiao\cite{xiao}. △ Less

Submitted 23 July, 2025; v1 submitted 2 June, 2025; originally announced June 2025.

An Exact System Optimum Assignment Model for Transit Demand Management

Authors: Xia Zhou, Mark Wallace, Daniel D. Harabor, Zhenliang Ma

Abstract: Mass transit systems are experiencing increasing congestion in many cities. The schedule-based transit assignment problem (STAP) involves a joint choice model for departure times and routes, defining a space-time path in which passengers decide when to depart and which route to take. User equilibrium (UE) models for the STAP indicates the current congestion cost, while a system optimum (SO) models… ▽ More Mass transit systems are experiencing increasing congestion in many cities. The schedule-based transit assignment problem (STAP) involves a joint choice model for departure times and routes, defining a space-time path in which passengers decide when to depart and which route to take. User equilibrium (UE) models for the STAP indicates the current congestion cost, while a system optimum (SO) models can provide insights for congestion relief directions. However, current STAP methods rely on approximate SO (Approx. SO) models, which underestimate the potential for congestion reduction in the system. The few studies in STAP that compute exact SO solutions ignore realistic constraints such as hard capacity, multi-line networks, or spatial-temporal competing demand flows. The paper proposes an exact SO method for the STAP that overcomes these limitations. We apply our approach to a case study involving part of the Hong Kong Mass Transit Railway network, which includes 5 lines, 12 interacting origin-destination pairs and 52,717 passengers. Computing an Approx. SO solution for this system indicates a modest potential for congestion reduction measures, with a cost reduction of 17.39% from the UE solution. Our exact SO solution is 36.35% lower than the UE solution, which is more than double the potential for congestion reduction. We then show how the exact SO solution can be used to identify opportunities for congestion reduction: (i) which origin-destination pairs have the most potential to reduce congestion; (ii) how many passengers can be reasonably shifted; (iii) future system potential with increasing demand and expanding network capacity. △ Less

Submitted 28 May, 2025; originally announced May 2025.

Comments: 18 pages, 13 figures

Uniqueness and nonuniqueness of $p$-harmonic Green functions on weighted $\mathbf{R}^n$ and metric spaces

Authors: Anders Björn, Jana Björn, Sylvester Eriksson-Bique, Xiaodan Zhou

Abstract: We study uniqueness of $p$-harmonic Green functions in domains $Ω$ in a complete metric space equipped with a doubling measure supporting a $p$-Poincaré inequality, with $1<p<\infty$. For bounded domains in unweighted $\mathbf{R}^n$, the uniqueness was shown for the $p$-Laplace operator $Δ_p$ and all $p$ by Kichenassamy--Véron (Math. Ann. 275 (1986), 599-615), while for $p=2$ it is an easy consequ… ▽ More We study uniqueness of $p$-harmonic Green functions in domains $Ω$ in a complete metric space equipped with a doubling measure supporting a $p$-Poincaré inequality, with $1<p<\infty$. For bounded domains in unweighted $\mathbf{R}^n$, the uniqueness was shown for the $p$-Laplace operator $Δ_p$ and all $p$ by Kichenassamy--Véron (Math. Ann. 275 (1986), 599-615), while for $p=2$ it is an easy consequence of the linearity of the Laplace operator $Δ$. Beyond that, uniqueness is only known in some particular cases, such as in Ahlfors $p$-regular spaces, as shown by Bonk--Capogna--Zhou (arXiv:2211.11974). When the singularity $x_0$ has positive $p$-capacity, the Green function is a particular multiple of the capacitary potential for $\text{cap}_p(\{x_0\},Ω)$ and is therefore unique. Here we give a sufficient condition for uniqueness in metric spaces, and provide an example showing that the range of $p$ for which it holds (while $x_0$ has zero $p$-capacity) can be a nondegenerate interval. In the opposite direction, we give the first example showing that uniqueness can fail in metric spaces, even for $p=2$. △ Less

Submitted 25 May, 2025; originally announced May 2025.

MSC Class: Primary: 35J08; Secondary: 30L99; 31C45; 31E05; 35J92; 49Q20

Departure time choice user equilibrium for public transport demand management

Authors: Xia Zhou, Zhenliang Ma, Mark Wallace, Daniel D. Harabor

Abstract: Departure time management is an efficient way in addressing the peak-hour crowding in public transport by reducing the temporal imbalance between service supply and travel demand. From the demand management perspective, the problem is to determine an equilibrium distribution of departure times for which no user can reduce their generalized cost by changing their departure times unilaterally. This… ▽ More Departure time management is an efficient way in addressing the peak-hour crowding in public transport by reducing the temporal imbalance between service supply and travel demand. From the demand management perspective, the problem is to determine an equilibrium distribution of departure times for which no user can reduce their generalized cost by changing their departure times unilaterally. This study introduces the departure time choice user equilibrium problem in public transport (DTUE-PT) for multi-line, schedule-based networks with hard train capacity constraints. We model the DTUE-PT problem as a Non-linear Mathematical Program problem (NMP) (minimizing the system gap) with a simulation model describing the complex system dynamics and passenger interactions. We develop an efficient, adaptive gap-based descent direction (AdaGDD) solution algorithm to solve the NMP problem. We validate the methodology on a multi-line public transport network with transfers by comparing with classical public transport assignment benchmark models, including Method of Successive Average (MSA) and day-to-day learning methods. The results show that the model can achieve a system gap ratio (the solution gap relative to the ideal least cost of an origin-destination option) of 0.1926, which significantly improves the solution performance from day-to-day learning (85%) and MSA (76%) algorithms. The sensitivity analysis highlights the solution stability of AdaGDD method over initial solution settings. The potential use of DTUE-PT model is demonstrated for evaluating the network design of Hong Kong mass transit railway network and can be easily extended to incorporate the route choice. △ Less

Submitted 21 May, 2025; originally announced May 2025.

Comments: 16 pages, 10 figures, 5 tables

On the $m$th order $p$-affine capacity

Authors: Xia Zhou, Deping Ye

Abstract: Let $M_{n, m}(\mathbb{R})$ denote the space of $n\times m$ real matrices, and $\mathcal{K}_o^{n,m}$ be the set of convex bodies in $M_{n, m}(\mathbb{R})$ containing the origin. We develop a theory for the $m$th order $p$-affine capacity $C_{p,Q}(\cdot)$ for $p\in[1,n)$ and $Q\in\mathcal{K}_{o}^{1,m}$. Several equivalent definitions for the $m$th order $p$-affine capacity will be provided, and some… ▽ More Let $M_{n, m}(\mathbb{R})$ denote the space of $n\times m$ real matrices, and $\mathcal{K}_o^{n,m}$ be the set of convex bodies in $M_{n, m}(\mathbb{R})$ containing the origin. We develop a theory for the $m$th order $p$-affine capacity $C_{p,Q}(\cdot)$ for $p\in[1,n)$ and $Q\in\mathcal{K}_{o}^{1,m}$. Several equivalent definitions for the $m$th order $p$-affine capacity will be provided, and some of its fundamental properties will be proved, including for example, translation invariance and affine invariance. We also establish several inequalities related to the $m$th order $p$-affine capacity, including those comparing to the $p$-variational capacity, the volume, the $m$th order $p$-integral affine surface area, as well as the $L_p$ surface area. △ Less

Submitted 18 May, 2025; originally announced May 2025.

MSC Class: 52A40; 52A38; 53A15; 46E30; 46E35; 28A75

Enhanced Error-free Retrieval in Kuramoto-type Associative-memory Networks via Two-memory Configuration

Authors: Zhuchun Li, Xiaoxue Zhao, Xiang Zhou

Abstract: We study the associative-memory network of Kuramoto-type oscillators that stores a set of memorized patterns (memories). In [Phys. Rev. Lett., 92 (2004), 108101], Nishikawa, Lai and Hoppensteadt showed that the capacity of this system for pattern retrieval with small errors can be made as high as that of the Hopfield network. Some stability analysis efforts focus on mutually orthogonal memories; h… ▽ More We study the associative-memory network of Kuramoto-type oscillators that stores a set of memorized patterns (memories). In [Phys. Rev. Lett., 92 (2004), 108101], Nishikawa, Lai and Hoppensteadt showed that the capacity of this system for pattern retrieval with small errors can be made as high as that of the Hopfield network. Some stability analysis efforts focus on mutually orthogonal memories; however, the theoretical results do not ensure error-free retrieval in general situations. In this paper, we present a route for using the model in pattern retrieval problems with small or large errors. We employ the eigenspectrum analysis of Jacobians and potential analysis of the gradient flow to derive the stability/instability of binary patterns. For two memories, the eigenspectrum of Jacobian at each pattern can be specified, which enables us to give the critical value of the parameter to distinguish the memories from all other patterns in stability. This setting of two memories substantially reduces the number of stable patterns and enlarges their basins, allowing us to recover defective patterns. We extend this approach to general cases and present a deterministic method for ensuring error-free retrieval across a general set of standard patterns. Numerical simulations and comparative analyses illustrate the approach. △ Less

Submitted 17 May, 2025; originally announced May 2025.

Comments: 23 pages, 5 figures, 1 table

MSC Class: 34C15; 92C42

Super-fast rates of convergence for Neural Networks Classifiers under the Hard Margin Condition

Authors: Nathanael Tepakbong, Ding-Xuan Zhou, Xiang Zhou

Abstract: We study the classical binary classification problem for hypothesis spaces of Deep Neural Networks (DNNs) with ReLU activation under Tsybakov's low-noise condition with exponent $q>0$, and its limit-case $q\to\infty$ which we refer to as the "hard-margin condition". We show that DNNs which minimize the empirical risk with square loss surrogate and $\ell_p$ penalty can achieve finite-sample excess… ▽ More We study the classical binary classification problem for hypothesis spaces of Deep Neural Networks (DNNs) with ReLU activation under Tsybakov's low-noise condition with exponent $q>0$, and its limit-case $q\to\infty$ which we refer to as the "hard-margin condition". We show that DNNs which minimize the empirical risk with square loss surrogate and $\ell_p$ penalty can achieve finite-sample excess risk bounds of order $\mathcal{O}\left(n^{-α}\right)$ for arbitrarily large $α>0$ under the hard-margin condition, provided that the regression function $η$ is sufficiently smooth. The proof relies on a novel decomposition of the excess risk which might be of independent interest. △ Less

Submitted 13 May, 2025; originally announced May 2025.

Comments: 31 pages

Normalized solutions for nonhomogeneous Chern-Simons-Schrödinger equations with critical exponential growth

Authors: Chenlu Wei, Sitong Chen, Xinao Zhou

Abstract: This paper investigates the existence of normalized solutions for the following Chern-Simons-Schrödinger equation: \begin{equation*} \left\{ \begin{array}{ll} -Δu+λu+\left(\frac{h^{2}(\vert x\vert)}{\vert x\vert^{2}}+\int_{\vert x\vert}^{\infty}\frac{h(s)}{s}u^{2}(s)\mathrm{d}s\right)u =\left(e^{u^2}-1\right)u+g(x), & x\in \R^2, u\in H_r^1(\R^2),\ \int_{\R^2}u^2\mathrm{d}x=c, \end{arra… ▽ More This paper investigates the existence of normalized solutions for the following Chern-Simons-Schrödinger equation: \begin{equation*} \left\{ \begin{array}{ll} -Δu+λu+\left(\frac{h^{2}(\vert x\vert)}{\vert x\vert^{2}}+\int_{\vert x\vert}^{\infty}\frac{h(s)}{s}u^{2}(s)\mathrm{d}s\right)u =\left(e^{u^2}-1\right)u+g(x), & x\in \R^2, u\in H_r^1(\R^2),\ \int_{\R^2}u^2\mathrm{d}x=c, \end{array} \right. \end{equation*} where $c>0$, $λ\in \R$ acts as a Lagrange multiplier and $g\in \mathcal {C}(\mathbb{R}^2,[0,+\infty))$ satisfies suitable assumptions. In addition to the loss of compactness caused by the nonlinearity with critical exponential growth, the intricate interactions among it, the nonlocal term, and the nonhomogeneous term significantly affect the geometric structure of the constrained functional, thereby making this research particularly challenging. By specifying explicit conditions on $c$, we subtly establish a structure of local minima of the constrained functional. Based on the structure, we employ new analytical techniques to prove the existence of two solutions: one being a local minimizer and one of mountain-pass type. Our results are entirely new, even for the Schrödinger equation that is when nonlocal terms are absent. We believe our methods may be adapted and modified to deal with more constrained problems with nonhomogeneous perturbation. △ Less

Submitted 29 April, 2025; originally announced April 2025.

Efficient state transition algorithm with guaranteed optimality

Authors: Xiaojun Zhou, Chunhua Yang, Weihua Gui

Abstract: As a constructivism-based intelligent optimization method, state transition algorithm (STA) has exhibited powerful search ability in optimization. However, the standard STA still shows slow convergence at a later stage for flat landscape and a user has to preset its maximum number of iterations (or function evaluations) by experience. To resolve these two issues, efficient state transition algorit… ▽ More As a constructivism-based intelligent optimization method, state transition algorithm (STA) has exhibited powerful search ability in optimization. However, the standard STA still shows slow convergence at a later stage for flat landscape and a user has to preset its maximum number of iterations (or function evaluations) by experience. To resolve these two issues, efficient state transition algorithm is proposed with guaranteed optimality. Firstly, novel translation transformations based on predictive modeling are proposed to generate more potential candidates by utilizing historical information. Secondly, parameter control strategies are proposed to accelerate the convergence. Thirdly, a specific termination condition is designed to guarantee that the STA can stop automatically at an optimal point, which is equivalent to the zero gradient in mathematical programming. Experimental results have demonstrated the effectiveness and superiority of the proposed method. △ Less

Submitted 19 April, 2025; originally announced April 2025.

Comments: 13 pages

MSC Class: 90 ACM Class: I.2

Structure of some mapping spaces

Authors: Liangzhao Zhang, Xiangyu Zhou

Abstract: We prove that the path space of a differentiable manifold is diffeomorphic to a Fréchet space, endowing the path space with a linear structure. Furthermore, the base point preserving mapping space consisting of maps from a cube to a differentiable manifold is also diffeomorphic to a Fréchet space. As a corollary of a more general theorem, we prove that the path fibration becomes a fibre bundle for… ▽ More We prove that the path space of a differentiable manifold is diffeomorphic to a Fréchet space, endowing the path space with a linear structure. Furthermore, the base point preserving mapping space consisting of maps from a cube to a differentiable manifold is also diffeomorphic to a Fréchet space. As a corollary of a more general theorem, we prove that the path fibration becomes a fibre bundle for manifolds M. Additionally, we discuss the mapping space from a compact topological space to a differentiable manifold, demonstrating that this space admits the structure of a smooth Banach manifold. △ Less

Submitted 15 April, 2025; originally announced April 2025.

Comments: 32 pages, comments are welcome

Boundary behavior at infinity for simple exchangeable fragmentation-coagulation process in critical slow regime

Authors: Lina Ji, Xiaowen Zhou

Abstract: For a critical simple exchangeable fragmentation-coagulation process in slow regime where the coagulation rate and fragmentation rate are of the same order, we show that there exist phase transitions for its boundary behavior at infinity depending on the asymptotics of the difference between the two rates, and find rather sharp conditions for different boundary behaviors. For a critical simple exchangeable fragmentation-coagulation process in slow regime where the coagulation rate and fragmentation rate are of the same order, we show that there exist phase transitions for its boundary behavior at infinity depending on the asymptotics of the difference between the two rates, and find rather sharp conditions for different boundary behaviors. △ Less

Submitted 7 May, 2025; v1 submitted 5 April, 2025; originally announced April 2025.

The non-abelian extension and Wells map of Leibniz conformal algebra

Authors: Jun Zhao, Bo Hou, Xin Zhou

Abstract: In this paper, we study the theory of non-abelian extensions of a Leibniz conformal algebra $R$ by a Leibniz conformal algebra $H$ and prove that all the non-abelian extensions are classified by non-abelian $2$nd cohomology $H^2_{nab}(R,H)$ in the sense of equivalence. Then we introduce a differential graded Lie algebra $\mathfrak{L}$ and show that the set of its Maurer-Cartan elements in bijectio… ▽ More In this paper, we study the theory of non-abelian extensions of a Leibniz conformal algebra $R$ by a Leibniz conformal algebra $H$ and prove that all the non-abelian extensions are classified by non-abelian $2$nd cohomology $H^2_{nab}(R,H)$ in the sense of equivalence. Then we introduce a differential graded Lie algebra $\mathfrak{L}$ and show that the set of its Maurer-Cartan elements in bijection with the set of non-abelian extensions. Finally, as an application of non-abelian extension, we consider the inducibility of a pair of automorphisms about a non-abelian extension, and give the fundamental sequence of Wells of Leibniz conformal algebra $R$. Especially, we discuss the extensibility problem of derivations about an abelian extension of $R$. △ Less

Submitted 31 March, 2025; v1 submitted 20 March, 2025; originally announced March 2025.

Single-Impulse Reachable Set in Arbitrary Dynamics Using Polynomials

Authors: Xingyu Zhou, Roberto Armellin, Dong Qiao, Xiangyu Li

Abstract: This paper presents a method to determine the reachable set (RS) of spacecraft after a single velocity impulse with an arbitrary direction, which is appropriate for the RS in both the state and observation spaces under arbitrary dynamics, extending the applications of current RS methods from two-body to arbitrary dynamics. First, the single-impulse RS model is generalized as a family of two-variab… ▽ More This paper presents a method to determine the reachable set (RS) of spacecraft after a single velocity impulse with an arbitrary direction, which is appropriate for the RS in both the state and observation spaces under arbitrary dynamics, extending the applications of current RS methods from two-body to arbitrary dynamics. First, the single-impulse RS model is generalized as a family of two-variable parameterized polynomials in the differential algebra scheme. Then, using the envelope theory, the boundary of RS is identified by solving the envelope equation. A framework is proposed to reduce the complexity of solving the envelope equation by converting it to the problem of searching the root of a one-variable polynomial. Moreover, a high-order local polynomial approximation for the RS envelope is derived to improve computational efficiency. The method successfully determines the RSs of two near-rectilinear halo orbits in the cislunar space. Simulation results show that the RSs in both state and observation spaces can be accurately approximated under the three-body dynamics, with relative errors of less than 0.0658%. In addition, using the local polynomial approximation, the computational time for solving the envelope equation is reduced by more than 84%. △ Less

Submitted 16 February, 2025; originally announced February 2025.

Log truncated threshold and zero mass conjecture

Authors: Fusheng Deng, Yinji Li, Qunhuan Liu, Zhiwei Wang, Xiangyu Zhou

Abstract: For plurisubharmonic functions $\varphi$ and $ψ$ lying in the Cegrell class of $\mathbb{B}^n$ and $\mathbb{B}^m$ respectively such that the Lelong number of $\varphi$ at the origin vanishes, we show that the mass of the origin with respect to the measure $(dd^c\max\{\varphi(z), ψ(Az)\})^n$ on $\mathbb{C}^n$ is zero for $A\in \mbox{Hom}(\mathbb{C}^n,\mathbb{C}^m)=\mathbb{C}^{nm}$ outside a pluripol… ▽ More For plurisubharmonic functions $\varphi$ and $ψ$ lying in the Cegrell class of $\mathbb{B}^n$ and $\mathbb{B}^m$ respectively such that the Lelong number of $\varphi$ at the origin vanishes, we show that the mass of the origin with respect to the measure $(dd^c\max\{\varphi(z), ψ(Az)\})^n$ on $\mathbb{C}^n$ is zero for $A\in \mbox{Hom}(\mathbb{C}^n,\mathbb{C}^m)=\mathbb{C}^{nm}$ outside a pluripolar set. For a plurisubharmonic function $\varphi$ near the origin in $\mathbb{C}^n$, we introduce a new concept coined the log truncated threshold of $\varphi$ at $0$ which reflects a singular property of $\varphi$ via a log function near the origin (denoted by $lt(\varphi,0)$) and derive an optimal estimate of the residual Monge-Ampère mass of $\varphi$ at $0$ in terms of its higher order Lelong numbers $ν_j(\varphi)$ at $0$ for $1\leq j\leq n-1$, in the case that $lt(\varphi,0)<\infty$. These results provide a new approach to the zero mass conjecture of Guedj and Rashkovskii, and unify and strengthen well-known results about this conjecture. △ Less

Submitted 27 January, 2025; originally announced January 2025.

Comments: Comments are very welcome!

MSC Class: 32U05; 32U25; 32U40; 32W20; 31C10

Iterative Proximal-Minimization for Computing Saddle Points with Fixed Index

Authors: Shuting Gu, Hao Zhang, Xiaoqun Zhang, Xiang Zhou

Abstract: Computing saddle points with a prescribed Morse index on potential energy surfaces is crucial for characterizing transition states for nosie-induced rare transition events in physics and chemistry. Many numerical algorithms for this type of saddle points are based on the eigenvector-following idea and can be cast as an iterative minimization formulation (SINUM. Vol. 53, p.1786, 2015), but they may… ▽ More Computing saddle points with a prescribed Morse index on potential energy surfaces is crucial for characterizing transition states for nosie-induced rare transition events in physics and chemistry. Many numerical algorithms for this type of saddle points are based on the eigenvector-following idea and can be cast as an iterative minimization formulation (SINUM. Vol. 53, p.1786, 2015), but they may struggle with convergence issues and require good initial guesses. To address this challenge, we discuss the differential game interpretation of this iterative minimization formulation and investigate the relationship between this game's Nash equilibrium and saddle points on the potential energy surface. Our main contribution is that adding a proximal term, which grows faster than quadratic, to the game's cost function can enhance the stability and robustness. This approach produces a robust Iterative Proximal Minimization (IPM) algorithm for saddle point computing. We show that the IPM algorithm surpasses the preceding methods in robustness without compromising the convergence rate or increasing computational expense. The algorithm's efficacy and robustness are showcased through a two-dimensional test problem, and the Allen-Cahn, Cahn-Hilliard equation, underscoring its numerical robustness. △ Less

Submitted 24 January, 2025; originally announced January 2025.

Comments: arXiv admin note: text overlap with arXiv:2212.08256

The $m$th order Orlicz projection bodies

Authors: Xia Zhou, Deping Ye, Zengle Zhang

Abstract: Let $M_{n, m}(\mathbb{R})$ be the space of $n\times m$ real matrices. Define $\mathcal{K}_o^{n,m}$ as the set of convex compact subsets in $M_{n,m}(\mathbb{R})$ with nonempty interior containing the origin $o\in M_{n, m}(\mathbb{R})$, and $\mathcal{K}_{(o)}^{n,m}$ as the members of $\mathcal{K}_o^{n,m}$ containing $o$ in their interiors. Let $Φ: M_{1, m}(\mathbb{R}) \rightarrow [0, \infty)$ be a c… ▽ More Let $M_{n, m}(\mathbb{R})$ be the space of $n\times m$ real matrices. Define $\mathcal{K}_o^{n,m}$ as the set of convex compact subsets in $M_{n,m}(\mathbb{R})$ with nonempty interior containing the origin $o\in M_{n, m}(\mathbb{R})$, and $\mathcal{K}_{(o)}^{n,m}$ as the members of $\mathcal{K}_o^{n,m}$ containing $o$ in their interiors. Let $Φ: M_{1, m}(\mathbb{R}) \rightarrow [0, \infty)$ be a convex function such that $Φ(o)=0$ and $Φ(z)+Φ(-z)>0$ for $z\neq o.$ In this paper, we propose the $m$th order Orlicz projection operator $Π_Φ^m: \mathcal{K}_{(o)}^{n,1}\rightarrow \mathcal{K}_{(o)}^{n,m}$, and study its fundamental properties, including the continuity and affine invariance. We establish the related higher-order Orlicz-Petty projection inequality, which states that the volume of $Π_Φ^{m, *}(K)$, the polar body of $Π_Φ^{m}(K)$, is maximized at origin-symmetric ellipsoids among convex bodies with fixed volume. Furthermore, when $Φ$ is strictly convex, we prove that the maximum is uniquely attained at origin-symmetric ellipsoids. Our proof is based on the classical Steiner symmetrization and its higher-order analogue. We also investigate the special case for $Φ_{Q}=φ\circ h_Q$, where $h_Q$ denotes the support function of $Q\in \mathcal{K}^{1, m}_o$ and $φ: [0, \infty)\rightarrow [0, \infty)$ is a convex function such that $φ(0)=0$ and $φ$ is strictly increasing on $[0, \infty).$ We establish a higher-order Orlicz-Petty projection inequality related to $Π_{Φ_Q}^{m, *} (K)$. Although $Φ_Q$ may not be strictly convex, we characterize the equality under the additional assumption on $Q$ and $φ$, such as $Q\in \mathcal{K}_{(o)}^{1,m}$ and the strict convexity of $φ$. △ Less

Submitted 23 June, 2025; v1 submitted 13 January, 2025; originally announced January 2025.

MSC Class: 52A39; 52A40; Secondary: 28A75

CeViT: Copula-Enhanced Vision Transformer in multi-task learning and bi-group image covariates with an application to myopia screening

Authors: Chong Zhong, Yang Li, Jinfeng Xu, Xiang Fu, Yunhao Liu, Qiuyi Huang, Danjuan Yang, Meiyan Li, Aiyi Liu, Alan H. Welsh, Xingtao Zhou, Bo Fu, Catherine C. Liu

Abstract: We aim to assist image-based myopia screening by resolving two longstanding problems, "how to integrate the information of ocular images of a pair of eyes" and "how to incorporate the inherent dependence among high-myopia status and axial length for both eyes." The classification-regression task is modeled as a novel 4-dimensional muti-response regression, where discrete responses are allowed, tha… ▽ More We aim to assist image-based myopia screening by resolving two longstanding problems, "how to integrate the information of ocular images of a pair of eyes" and "how to incorporate the inherent dependence among high-myopia status and axial length for both eyes." The classification-regression task is modeled as a novel 4-dimensional muti-response regression, where discrete responses are allowed, that relates to two dependent 3rd-order tensors (3D ultrawide-field fundus images). We present a Vision Transformer-based bi-channel architecture, named CeViT, where the common features of a pair of eyes are extracted via a shared Transformer encoder, and the interocular asymmetries are modeled through separated multilayer perceptron heads. Statistically, we model the conditional dependence among mixture of discrete-continuous responses given the image covariates by a so-called copula loss. We establish a new theoretical framework regarding fine-tuning on CeViT based on latent representations, allowing the black-box fine-tuning procedure interpretable and guaranteeing higher relative efficiency of fine-tuning weight estimation in the asymptotic setting. We apply CeViT to an annotated ultrawide-field fundus image dataset collected by Shanghai Eye \& ENT Hospital, demonstrating that CeViT enhances the baseline model in both accuracy of classifying high-myopia and prediction of AL on both eyes. △ Less

Submitted 11 January, 2025; originally announced January 2025.

Lévy Score Function and Score-Based Particle Algorithm for Nonlinear Lévy--Fokker--Planck Equations

Authors: Yuanfei Huang, Chengyu Liu, Xiang Zhou

Abstract: The score function for the diffusion process, also known as the gradient of the log-density, is a basic concept to characterize the probability flow with important applications in the score-based diffusion generative modelling and the simulation of Itô stochastic differential equations. However, neither the probability flow nor the corresponding score function for the diffusion-jump process are kn… ▽ More The score function for the diffusion process, also known as the gradient of the log-density, is a basic concept to characterize the probability flow with important applications in the score-based diffusion generative modelling and the simulation of Itô stochastic differential equations. However, neither the probability flow nor the corresponding score function for the diffusion-jump process are known. This paper delivers mathematical derivation, numerical algorithm, and error analysis focusing on the corresponding score function in non-Gaussian systems with jumps and discontinuities represented by the nonlinear Lévy--Fokker--Planck equations. We propose the Lévy score function for such stochastic equations, which features a nonlocal double-integral term, and we develop its training algorithm by minimizing the proposed loss function from samples. Based on the equivalence of the probability flow with deterministic dynamics, we develop a self-consistent score-based transport particle algorithm to sample the interactive Lévy stochastic process at discrete time grid points. We provide error bound for the Kullback--Leibler divergence between the numerical and true probability density functions by overcoming the nonlocal challenges in the Lévy score. The full error analysis with the Monte Carlo error and the time discretization error is furthermore established. To show the usefulness and efficiency of our approach, numerical examples from applications in biology and finance are tested. △ Less

Submitted 27 December, 2024; originally announced December 2024.

Towards Simple and Provable Parameter-Free Adaptive Gradient Methods

Authors: Yuanzhe Tao, Huizhuo Yuan, Xun Zhou, Yuan Cao, Quanquan Gu

Abstract: Optimization algorithms such as AdaGrad and Adam have significantly advanced the training of deep models by dynamically adjusting the learning rate during the optimization process. However, adhoc tuning of learning rates poses a challenge, leading to inefficiencies in practice. To address this issue, recent research has focused on developing "learning-rate-free" or "parameter-free" algorithms that… ▽ More Optimization algorithms such as AdaGrad and Adam have significantly advanced the training of deep models by dynamically adjusting the learning rate during the optimization process. However, adhoc tuning of learning rates poses a challenge, leading to inefficiencies in practice. To address this issue, recent research has focused on developing "learning-rate-free" or "parameter-free" algorithms that operate effectively without the need for learning rate tuning. Despite these efforts, existing parameter-free variants of AdaGrad and Adam tend to be overly complex and/or lack formal convergence guarantees. In this paper, we present AdaGrad++ and Adam++, novel and simple parameter-free variants of AdaGrad and Adam with convergence guarantees. We prove that AdaGrad++ achieves comparable convergence rates to AdaGrad in convex optimization without predefined learning rate assumptions. Similarly, Adam++ matches the convergence rate of Adam without relying on any conditions on the learning rates. Experimental results across various deep learning tasks validate the competitive performance of AdaGrad++ and Adam++. △ Less

Submitted 26 December, 2024; originally announced December 2024.

Comments: 34 pages, 16 figures, 3 tables

Discrete spectrum of probability measures for locally compact group actions

Authors: Zongrui Hu, Xiao Ma, Leiye Xu, Xiaomin Zhou

Abstract: In this paper, we investigate the discrete spectrum of probability measures for actions of locally compact groups. We establish that a probability measure has a discrete spectrum if and only if it has bounded measure-max-mean-complexity. As applications: 1) An invariant measure for a locally compact amenable group action has a discrete spectrum if and only if it has bounded mean-complexity along… ▽ More In this paper, we investigate the discrete spectrum of probability measures for actions of locally compact groups. We establish that a probability measure has a discrete spectrum if and only if it has bounded measure-max-mean-complexity. As applications: 1) An invariant measure for a locally compact amenable group action has a discrete spectrum if and only if it has bounded mean-complexity along Følner sequences; 2) An invariant measure for a locally compact amenable group action has a discrete spectrum if and only if it is mean equicontinuous along a tempered Følner sequence, or equicontinuous in the mean along a tempered Følner sequence. △ Less

Submitted 30 January, 2025; v1 submitted 23 December, 2024; originally announced December 2024.

MSC Class: 37A15; 37B05

Mean--Variance Portfolio Selection by Continuous-Time Reinforcement Learning: Algorithms, Regret Analysis, and Empirical Study

Authors: Yilie Huang, Yanwei Jia, Xun Yu Zhou

Abstract: We study continuous-time mean--variance portfolio selection in markets where stock prices are diffusion processes driven by observable factors that are also diffusion processes, yet the coefficients of these processes are unknown. Based on the recently developed reinforcement learning (RL) theory for diffusion processes, we present a general data-driven RL algorithm that learns the pre-committed i… ▽ More We study continuous-time mean--variance portfolio selection in markets where stock prices are diffusion processes driven by observable factors that are also diffusion processes, yet the coefficients of these processes are unknown. Based on the recently developed reinforcement learning (RL) theory for diffusion processes, we present a general data-driven RL algorithm that learns the pre-committed investment strategy directly without attempting to learn or estimate the market coefficients. For multi-stock Black--Scholes markets without factors, we further devise a baseline algorithm and prove its performance guarantee by deriving a sublinear regret bound in terms of the Sharpe ratio. For performance enhancement and practical implementation, we modify the baseline algorithm and carry out an extensive empirical study to compare its performance, in terms of a host of common metrics, with a large number of widely employed portfolio allocation strategies on S\&P 500 constituents. The results demonstrate that the proposed continuous-time RL strategy is consistently among the best, especially in a volatile bear market, and decisively outperforms the model-based continuous-time counterparts by significant margins. △ Less

Submitted 10 August, 2025; v1 submitted 8 December, 2024; originally announced December 2024.

Comments: 82 pages, 6 figures, 7 tables

MSC Class: 68T05; 91G10; 68Q25; 93E35; 93E20

Optimal Rates for Robust Stochastic Convex Optimization

Authors: Changyu Gao, Andrew Lowy, Xingyu Zhou, Stephen J. Wright

Abstract: Machine learning algorithms in high-dimensional settings are highly susceptible to the influence of even a small fraction of structured outliers, making robust optimization techniques essential. In particular, within the $ε$-contamination model, where an adversary can inspect and replace up to an $ε$-fraction of the samples, a fundamental open problem is determining the optimal rates for robust st… ▽ More Machine learning algorithms in high-dimensional settings are highly susceptible to the influence of even a small fraction of structured outliers, making robust optimization techniques essential. In particular, within the $ε$-contamination model, where an adversary can inspect and replace up to an $ε$-fraction of the samples, a fundamental open problem is determining the optimal rates for robust stochastic convex optimization (SCO) under such contamination. We develop novel algorithms that achieve minimax-optimal excess risk (up to logarithmic factors) under the $ε$-contamination model. Our approach improves over existing algorithms, which are not only suboptimal but also require stringent assumptions, including Lipschitz continuity and smoothness of individual sample functions. By contrast, our optimal algorithms do not require these stringent assumptions, assuming only population-level smoothness of the loss. Moreover, our algorithms can be adapted to handle the case in which the covariance parameter is unknown, and can be extended to nonsmooth population risks via convolutional smoothing. We complement our algorithmic developments with a tight information-theoretic lower bound for robust SCO. △ Less

Submitted 23 April, 2025; v1 submitted 14 December, 2024; originally announced December 2024.

Comments: The 6th annual Symposium on Foundations of Responsible Computing (FORC 2025)

Regularity of stochastic differential equations on the Wiener space by coupling

Authors: Stefan Geiss, Xilin Zhou

Abstract: Using the coupling method introduced in \cite{Geiss:Ylinen:21}, we investigate regularity properties of stochastic differential equations, where we consider the Lipschitz case in $\R^d$ and allow for Hölder continuity of the diffusion coefficient of scalar valued stochastic differential equations. Two cases of the coupling method are of special interest: The uniform coupling to treat the Malliavin… ▽ More Using the coupling method introduced in \cite{Geiss:Ylinen:21}, we investigate regularity properties of stochastic differential equations, where we consider the Lipschitz case in $\R^d$ and allow for Hölder continuity of the diffusion coefficient of scalar valued stochastic differential equations. Two cases of the coupling method are of special interest: The uniform coupling to treat the Malliavin Sobolev space $\D_{1,2}$ and real interpolation spaces, and secondly a cut-off coupling to treat the $L_p$-variation of backward stochastic differential equations where the forward process is the investigated stochastic differential equation. △ Less

Submitted 20 May, 2025; v1 submitted 14 December, 2024; originally announced December 2024.

Comments: Results added, improvement of presentation, and change of title to reflect better the content of the manuscript

MSC Class: 60H07; 60H10; 46E35; 46B70

A Novel Methodology in Credit Spread Prediction Based on Ensemble Learning and Feature Selection

Authors: Yu Shao, Jiawen Bai, Yingze Hou, Xia'an Zhou, Zhanhao Pan

Abstract: The credit spread is a key indicator in bond investments, offering valuable insights for fixed-income investors to devise effective trading strategies. This study proposes a novel credit spread forecasting model leveraging ensemble learning techniques. To enhance predictive accuracy, a feature selection method based on mutual information is incorporated. Empirical results demonstrate that the prop… ▽ More The credit spread is a key indicator in bond investments, offering valuable insights for fixed-income investors to devise effective trading strategies. This study proposes a novel credit spread forecasting model leveraging ensemble learning techniques. To enhance predictive accuracy, a feature selection method based on mutual information is incorporated. Empirical results demonstrate that the proposed methodology delivers superior accuracy in credit spread predictions. Additionally, we present a forecast of future credit spread trends using current data, providing actionable insights for investment decision-making. △ Less

Submitted 12 December, 2024; originally announced December 2024.

Comments: 7 pages, 5 figures

A priori estimates and moving plane method for a class of Grushin equation

Authors: Wolfram Bauer, Yawei Wei, Xiaodong Zhou

Abstract: In this paper, we study three kinds of nonlinear degenerate elliptic equations containing the Grushin operator. First, we prove radial symmetry and a decay rate at infinity of solutions to such a Grushin equation by using the moving plane method in combination with suitable integral inequalities. Applying similar methods, we obtain nonexistence results for solutions to a second type of Grushin equ… ▽ More In this paper, we study three kinds of nonlinear degenerate elliptic equations containing the Grushin operator. First, we prove radial symmetry and a decay rate at infinity of solutions to such a Grushin equation by using the moving plane method in combination with suitable integral inequalities. Applying similar methods, we obtain nonexistence results for solutions to a second type of Grushin equation in Euclidean space and in half space, respectively. Finally, we derive a priori estimates for positive solutions to more general types of Grushin equations by employing blow up analysis. △ Less

Submitted 10 December, 2024; originally announced December 2024.

MSC Class: 35J70; 35B45; 35A16

Extinction behaviour for a mutually interacting continuous-state population dynamics

Authors: Jie Xiong, Xu Yang, Xiaowen Zhou

Abstract: We consider a system of two stochastic differential equations (SDEs) with negative two-way interactions driven by Brownian motions and spectrally positive $α$-stable random measures. Such a SDE system can be identified as a Lotka-Volterra type population model. We find close to sharp conditions for one of the population to go extinct or extinguishing. We consider a system of two stochastic differential equations (SDEs) with negative two-way interactions driven by Brownian motions and spectrally positive $α$-stable random measures. Such a SDE system can be identified as a Lotka-Volterra type population model. We find close to sharp conditions for one of the population to go extinct or extinguishing. △ Less

Submitted 30 November, 2024; originally announced December 2024.

MARS: Unleashing the Power of Variance Reduction for Training Large Models

Authors: Huizhuo Yuan, Yifeng Liu, Shuang Wu, Xun Zhou, Quanquan Gu

Abstract: Training deep neural networks--and more recently, large models demands efficient and scalable optimizers. Adaptive gradient algorithms like Adam, AdamW, and their variants have been central to this task. Despite the development of numerous variance reduction algorithms in the past decade aimed at accelerating stochastic optimization in both convex and nonconvex settings, variance reduction has not… ▽ More Training deep neural networks--and more recently, large models demands efficient and scalable optimizers. Adaptive gradient algorithms like Adam, AdamW, and their variants have been central to this task. Despite the development of numerous variance reduction algorithms in the past decade aimed at accelerating stochastic optimization in both convex and nonconvex settings, variance reduction has not found widespread success in training deep neural networks or large language models. Consequently, it has remained a less favored approach in modern AI. In this paper, to unleash the power of variance reduction for efficient training of large models, we propose a unified optimization framework, MARS (Make vAriance Reduction Shine), which reconciles preconditioned gradient methods with variance reduction via a scaled stochastic recursive momentum technique. Within our framework, we introduce three instances of MARS that leverage preconditioned gradient updates based on AdamW, Lion, and Shampoo, respectively. We also draw a connection between our algorithms and existing optimizers. Experimental results on training GPT-2 models indicate that MARS consistently outperforms AdamW by a large margin. The implementation of MARS is available at https://github.com/AGI-Arena/MARS. △ Less

Submitted 4 September, 2025; v1 submitted 15 November, 2024; originally announced November 2024.

Comments: 35 pages, 19 figures, 12 tables

Two Kinds of Learning Algorithms for Continuous-Time VWAP Targeting Execution

Authors: Xingyu Zhou, Wenbin Chen, Mingyu Xu

Abstract: The optimal execution problem has always been a continuously focused research issue, and many reinforcement learning (RL) algorithms have been studied. In this article, we consider the execution problem of targeting the volume weighted average price (VWAP) and propose a relaxed stochastic optimization problem with an entropy regularizer to encourage more exploration. We derive the explicit formula… ▽ More The optimal execution problem has always been a continuously focused research issue, and many reinforcement learning (RL) algorithms have been studied. In this article, we consider the execution problem of targeting the volume weighted average price (VWAP) and propose a relaxed stochastic optimization problem with an entropy regularizer to encourage more exploration. We derive the explicit formula of the optimal policy, which is Gaussian distributed, with its mean value being the solution to the original problem. Extending the framework of continuous RL to processes with jumps, we provide some theoretical proofs for RL algorithms. First, minimizing the martingale loss function leads to the optimal parameter estimates in the mean-square sense, and the second algorithm is to use the martingale orthogonality condition. In addition to the RL algorithm, we also propose another learning algorithm: adaptive dynamic programming (ADP) algorithm, and verify the performance of both in two different environments across different random seeds. Convergence of all algorithms has been verified in different environments, and shows a larger advantage in the environment with stronger price impact. ADP is a good choice when the agent fully understands the environment and can estimate the parameters well. On the other hand, RL algorithms do not require any model assumptions or parameter estimation, and are able to learn directly from interactions with the environment. △ Less

Submitted 10 November, 2024; originally announced November 2024.

Regret of exploratory policy improvement and $q$-learning

Authors: Wenpin Tang, Xun Yu Zhou

Abstract: We study the convergence of $q$-learning and related algorithms introduced by Jia and Zhou (J. Mach. Learn. Res., 24 (2023), 161) for controlled diffusion processes. Under suitable conditions on the growth and regularity of the model parameters, we provide a quantitative error and regret analysis of both the exploratory policy improvement algorithm and the $q$-learning algorithm. We study the convergence of $q$-learning and related algorithms introduced by Jia and Zhou (J. Mach. Learn. Res., 24 (2023), 161) for controlled diffusion processes. Under suitable conditions on the growth and regularity of the model parameters, we provide a quantitative error and regret analysis of both the exploratory policy improvement algorithm and the $q$-learning algorithm. △ Less

Submitted 2 November, 2024; originally announced November 2024.

Comments: 23 pages, 1 figure

Non-dense orbits on topological dynamical systems

Authors: Cao Zhao, Jiao Yang, Xiaoyao Zhou

Abstract: Let $(X,d,T )$ be a topological dynamical system with the specification property. We consider the non-dense orbit set $E(z_0)$ and show that for any non-transitive point $z_0\in X$, this set $E(z_0)$ is empty or carries full topological pressure. Let $(X,d,T )$ be a topological dynamical system with the specification property. We consider the non-dense orbit set $E(z_0)$ and show that for any non-transitive point $z_0\in X$, this set $E(z_0)$ is empty or carries full topological pressure. △ Less

Submitted 7 October, 2024; originally announced October 2024.

New Approach for Interior Regularity of Monge-Ampère Equations

Authors: Ruosi Chen, Xingchen Zhou

Abstract: By developing an integral approach, we present a new method for the interior regularity of strictly convex solution of the Monge-Ampère equation $\det D^2 u = 1$. By developing an integral approach, we present a new method for the interior regularity of strictly convex solution of the Monge-Ampère equation $\det D^2 u = 1$. △ Less

Submitted 24 September, 2024; originally announced September 2024.

Model-Embedded Gaussian Process Regression for Parameter Estimation in Dynamical System

Authors: Ying Zhou, Jinglai Li, Xiang Zhou, Hongqiao Wang

Abstract: Identifying dynamical system (DS) is a vital task in science and engineering. Traditional methods require numerous calls to the DS solver, rendering likelihood-based or least-squares inference frameworks impractical. For efficient parameter inference, two state-of-the-art techniques are the kernel method for modeling and the "one-step framework" for jointly inferring unknown parameters and hyperpa… ▽ More Identifying dynamical system (DS) is a vital task in science and engineering. Traditional methods require numerous calls to the DS solver, rendering likelihood-based or least-squares inference frameworks impractical. For efficient parameter inference, two state-of-the-art techniques are the kernel method for modeling and the "one-step framework" for jointly inferring unknown parameters and hyperparameters. The kernel method is a quick and straightforward technique, but it cannot estimate solutions and their derivatives, which must strictly adhere to physical laws. We propose a model-embedded "one-step" Bayesian framework for joint inference of unknown parameters and hyperparameters by maximizing the marginal likelihood. This approach models the solution and its derivatives using Gaussian process regression (GPR), taking into account smoothness and continuity properties, and treats differential equations as constraints that can be naturally integrated into the Bayesian framework in the linear case. Additionally, we prove the convergence of the model-embedded Gaussian process regression (ME-GPR) for theoretical development. Motivated by Taylor expansion, we introduce a piecewise first-order linearization strategy to handle nonlinear dynamic systems. We derive estimates and confidence intervals, demonstrating that they exhibit low bias and good coverage properties for both simulated models and real data. △ Less

Submitted 18 September, 2024; originally announced September 2024.

Comments: 24 pages, 3 figures, 5 tables

MSC Class: 62F15

Realistic Extreme Behavior Generation for Improved AV Testing

Authors: Robert Dyro, Matthew Foutter, Ruolin Li, Luigi Di Lillo, Edward Schmerling, Xilin Zhou, Marco Pavone

Abstract: This work introduces a framework to diagnose the strengths and shortcomings of Autonomous Vehicle (AV) collision avoidance technology with synthetic yet realistic potential collision scenarios adapted from real-world, collision-free data. Our framework generates counterfactual collisions with diverse crash properties, e.g., crash angle and velocity, between an adversary and a target vehicle by add… ▽ More This work introduces a framework to diagnose the strengths and shortcomings of Autonomous Vehicle (AV) collision avoidance technology with synthetic yet realistic potential collision scenarios adapted from real-world, collision-free data. Our framework generates counterfactual collisions with diverse crash properties, e.g., crash angle and velocity, between an adversary and a target vehicle by adding perturbations to the adversary's predicted trajectory from a learned AV behavior model. Our main contribution is to ground these adversarial perturbations in realistic behavior as defined through the lens of data-alignment in the behavior model's parameter space. Then, we cluster these synthetic counterfactuals to identify plausible and representative collision scenarios to form the basis of a test suite for downstream AV system evaluation. We demonstrate our framework using two state-of-the-art behavior prediction models as sources of realistic adversarial perturbations, and show that our scenario clustering evokes interpretable failure modes from a baseline AV policy under evaluation. △ Less

Submitted 16 September, 2024; originally announced September 2024.